Downside Risk Neutral Probabilities DISCUSSION PAPER NO 756 DISCUSSION PAPER SERIES. April 2016

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1 ISSN Downside Risk Neutral Probabilities By Pierre Chaigneau Louis Eeckhoudt DISCUSSION PAPER NO 756 DISCUSSION PAPER SERIES April 06

2 Downside risk neutral probabilities Pierre Chaigneau HEC Montreal Louis Eeckhoudt Ieseg (Lille) and Core (Louvain) April 06 Abstract Risk neutral probabilities are adjusted to take into account the asset price effect of risk preferences. This paper introduces downside (respectively outer) risk neutral probabilities, which are adjusted to take into account the asset price effect of preferences for downside (resp. outer) risk and higher degree risks. Using risk preference theory, we interpret these three changes in probability measures in terms of risk substitution. With downside risk neutral probabilities, the pricing kernel is linear in wealth. Outer risk neutral probabilities can be viewed as a reasonable approximation of physical probabilities. Keywords: downside risk, pricing kernel, prudence, risk aversion, risk neutral probabilities, risk substitution. JEL codes: D8, G. We thank Georges Dionne, Christian Dorion, Mathieu Fournier, and Christian Gollier for useful comments and suggestions. Department of Finance, HEC Montreal, 3000 chemin de la cote Sainte Catherine, H3T AT Montreal, QC, Canada. pierre.chaigneau@hec.ca. Tel: Fax:

3 Risk preferences are not limited to risk aversion. Downside risk aversion, or prudence, is necessary for decreasing absolute risk aversion, for standard risk aversion (Kimball (993)), and it has been linked with precautionary savings (Kimball (990)). It is thus important to take into consideration this higher-order risk preference. However, the incorporation of risk preferences in economic models of decision under risk often leads to a loss in tractability. In addition, it is often hard to distinguish between the effect of preferences for different degrees of risk. For example, recent studies have shown that risk aversion and downside risk aversion both matter in asset pricing. But how is downside risk aversion incorporated into asset pricing formulas? The literature on risk preferences has identified several degrees of risk. In an expected utility framework with utility function u, the preference for each degree of risk is associated with a derivative of u of a different order. An increase in second degree risk, also known as an increase in risk, is an increase in the dispersion of a distribution in the sense of mean-preserving spreads. A risk averse agent, with u ll < 0, is averse to such increases. An increase in third degree risk, also known as an increase in downside risk, is a transfer of risk to the left of the distribution which leaves the mean and variance unchanged. A downside risk averse agent, with u lll > 0, is averse to such increases (Menezes, Geiss, and Tressler (980)). An increase in downside risk implies a lower third moment of the distribution, i.e., a lower skewness (Menezes, Geiss, and Tressler (980), Chiu (005)). An increase in fourth degree risk, also known as an increase in outer risk, is a transfer of risk from the center toward the tails which leaves the mean, variance, and skewness unchanged. An outer risk averse agent, with u (4) < 0, is averse to such increases (Menezes and Wang (005)). An increase in outer risk implies a higher fourth moment of the distribution, i.e., a higher kurtosis. These higher-order risk preferences can thus explain why especially the skewness but also the kurtosis of returns are determinants of expected returns (e.g., Harvey and Siddique (000), Dittmar (00)). There are different ways to incorporate these risk preferences into an asset pricing formula. Risk neutral probabilities allow to price assets as if investors were risk neutral. Specifically, if investors were risk neutral (with u ll = 0), then risk neutral probabilities would coincide with physical probabilities. If investors are not risk neutral (u ll < 0 if they are risk averse), then risk neutral probabilities are adjusted to take into account the asset price effect of risk aversion and The signs of successive derivatives of the utility function can be directly related to the preferences for successive degrees of risk, but they cannot be directly related to preferences for moments of the distribution. Indeed, stochastic dominance criteria are related to degrees of risk rather than to moments of the distribution. For example, considering two distributions A and B with the same mean, distribution B is dominated in a secondorder stochastic dominance sense if and only if it can be constructed by applying a sequence of mean-preserving spreads to distribution A (e.g. Gollier (00)). Moreover, a change in a degree of risk implies a certain change in the corresponding moment of the distribution, but the opposite is not necessarily true. For example, a meanpreserving spread implies a higher variance, but a higher variance does not imply a mean-preserving spread of the distribution. That is, if two distributions have the same mean but different variances, the distribution with the lower variance will not necessarily be preferred by a risk averse agent. Thus, to study the effect of risk preferences on asset prices in an expected utility framework, we work with degrees of risk rather than with moments. 3

4 higher-order risk attitudes. This paper makes three contributions. First, we introduce the concepts of downside and outer risk neutral probabilities, which are natural extensions of the concept of risk neutral probabilities. Second, we provide new asset pricing formulas based on these new probability measures. These formulas clarify the effects of risk aversion, downside risk aversion, and prudence on asset prices. Third, we provide interpretations of these changes in probability measures, as well as for the change in probability measure which yields risk neutral probabilities, in terms of risk substitution. Downside risk neutral probabilities allow to price assets as if investors were averse to risk, but were neutral with respect to higher degree risks, including downside risk and outer risk. Specifically, if investors were risk averse but downside risk neutral (with u ll < 0 and u lll = 0), then downside risk neutral probabilities would coincide with physical probabilities. If investors are not downside risk neutral (u lll > 0 if they are downside risk averse), then downside risk neutral probabilities are adjusted to take into account the asset price effect of downside risk aversion and higher-order risk attitudes. Due to this change in probability measure, the asset pricing formulas with downside risk neutral probabilities do not directly involve the (nonlinear) utility function. Instead, they incorporate the coefficient of absolute risk aversion evaluated at the initial level of wealth. This aspect is reminiscent of the Arrow-Pratt approximation of the risk premium, which is widely used due to its simplicity and its intuitive appeal. Yet the asset pricing formulas that involve downside risk neutral (or outer risk neutral) probabilities that we derive in this paper are not approximations: they yield the same asset prices as other asset pricing formulas. Outer risk neutral probabilities allow to price assets as if investors were risk averse and downside risk averse, but were neutral with respect to higher degree risks, including outer risk. Specifically, if investors were prudent but outer risk neutral (with u lll > 0 and u (4) = 0), then outer risk neutral probabilities would coincide with physical probabilities. If investors are not outer risk neutral (u (4) < 0 if they are outer risk averse), then outer risk neutral probabilities are adjusted to take into account the asset price effect of outer risk aversion and higher-order risk attitudes. Due to this change in probability measure, the asset pricing formulas with outer risk neutral probabilities do not directly involve the (nonlinear) utility function. Instead, they incorporate the coefficients of absolute risk aversion and of downside risk aversion evaluated at the initial level of wealth. We provide interpretations of the changes in probability measures in terms of risk substitution. This sheds light on the change in measure that yields the new probability measures introduced in this paper, but also on the change in measure that yields the well-known and widely used risk The change in probability measure is based on a first order Taylor expansion of marginal utility. Other papers have already used a Taylor expansion of marginal utility for asset pricing purposes. Harvey and Siddique (000), Dittmar (00), and Chabi-Yo (0), among others, approximate the pricing kernel with Taylor expansions of marginal utility of order two (respectively three). Downside risk aversion and outer risk aversion, also known respectively as prudence and temperance, have been defined as preferences over lotteries in Eeckhoudt and Schlesinger (006). 4

5 neutral probabilities. The change in probability measure which yields risk neutral probabilities is such that the first degree risk adjusts to incorporate the asset price effect of the preferences for the second and higher degree risks. This depends on the investor s rates of substitution between the second and higher degree risks on the one hand, and first degree risk on the other hand. In particular, the rate of substitution between second degree risk and first degree risk is the coefficient of absolute risk aversion, while the rate of substitution between third degree risk and first degree risk is the coefficient of downside risk aversion. The formulas in this paper show how risk neutral probabilities are adjusted to take into account the preferences for risk, but also the preferences for downside risk and for higher degree risks. The change in probability measure which yields downside risk neutral probabilities is such that the first two degree risks adjust to incorporate the asset price effect of the preferences for the third and higher degree risks. This depends on the investor s rates of substitution between the third and higher degree risks on the one hand, and the first and second degree risks on the other hand. In particular, the rate of substitution between third degree risk and first degree risk is the coefficient of downside risk aversion, while the rate of substitution between third degree risk and second degree risk is the coefficient of absolute prudence. Finally, the change in probability measure which yields outer risk neutral probabilities is such that the first three degree risks adjust to incorporate the asset price effect of the preferences for the fourth and higher degree risks. This depends on the investor s rates of substitution between the fourth and higher degree risks on the one hand, and the first, second and third degree risks on the other hand. The downside risk neutral probability measure can improve tractability in asset pricing models. Indeed, the pricing kernel associated with downside risk neutral probabilities is linear in future wealth. This sets our analysis apart from a number of recent papers, such as Eraker (008) and Martin (03), which also derive new analytical expressions for asset prices, in which the pricing kernel is not linear in state variables. These papers use physical probabilities and as- sume CRRA or Epstein-Zin preferences, whereas we use new probability measures and we make minimal assumptions on the utility function. Linearity in state variables is advantageous for analytical tractability, interpretation, and empirical implementation (e.g., Brandt and Chapman (04)). A linear pricing kernel also allows to use the Capital Asset Pricing Model (CAPM) to price assets, without making strong assumptions on risk preferences or probability distributions. By contrast, with physical probabilities, a simple CAPM-like formula requires strong and potentially unrealistic assumptions. It is possible to incorporate the asset pricing effect of higher-order risk preferences with physical probabilities (e.g., equation (7a) in Harvey and Siddique (000)). However, the formula then involves a number of additional terms, and it only holds as an ap- proximation with a finite number of additional terms (unless the utility function is neutral with respect to some higher degree risk). With downside risk neutral probabilities, a simple CAPMlike formula holds with minimal assumptions, because the asset pricing effect of higher-order risk preferences is incorporated as a probability adjustment, and it is not an approximation. 5

6 The outer risk neutral probability measure leads to asset pricing formulas which are less tractable than under the risk neutral or the downside risk neutral probability measure, but more tractable than under the physical probability measure. Interestingly, the outer risk neutral measure could be a close approximation of the physical probability measure. Indeed, outer risk neutral probabilities coincide with physical probabilities if and only if the utility function is outer risk neutral (whether or not it is downside risk averse), a case which is not inconsistent with empirical findings on risk preferences (Deck and Schlesinger (00)). When in addition the utility function is downside risk averse, we show that the risk neutral distribution has fat tails relative to the physical distribution, and the pricing kernel is U-shaped, consistent with the empirical evidence. In this case, we also relate the level of wealth at which the pricing kernel reaches its minimum value to the coefficient of absolute prudence. The model For simplicity, we consider an economy with two dates: t = 0 and t = (in Appendix C, we consider an economy with T future dates). Current aggregate wealth in the economy is w 0, and future aggregate wealth is w. It is equal to w s in state of the world s, for s {,..., S}, which occurs with probability p s 0, with S p s= s =. There is a representative agent, an expected utility maximizer with time separable preferences. They are described by a subjective discount factor β (0, ], and a utility function u such that u l > 0. In addition, u ll < 0 if the agent is risk averse, u lll > 0 if the agent is downside risk averse, and u (4) < 0 if the agent is outer risk averse (see Scott and Horvath (980)). For example, CARA and CRRA utility functions are risk averse, downside risk averse, and outer risk averse. The payoff of an asset at time t = is a random variable x, which is equal to x s in state of the world s. The values of w 0, w s and x s are finite for any s. Using the standard stochastic discount factor formula (e.g. Hansen and Jagannathan (99)), the price at t = 0 of any given asset with payoff x is where β ut (w ) u t (w ) 0 P = E β u l (w ) x u l (w 0 ), () is the stochastic discount factor or pricing kernel, and E[ ] is the expectation operator with respect to the physical probability measure, using the information at time 0. 6

7 Risk neutral probabilities. Definition and properties Assume that u is of class C. For any given s, let η,s be defined implicitly as u l (w s ) η,s u l (w 0 ). () Definition Let ν S s= p s η,s, and λ,s p sη,s ν = S p s η,s s= p s η,s. (3) The set {λ,s } is the set of risk neutral probabilities, and Λ is the risk neutral probability measure. By construction, S λ,s =. Note that dλ,s s= dp s ν = η,s is the Radon-Nikodym derivative of the risk neutral measure with respect to the physical measure. With linear utility, the risk neutral probability measure coincides with the physical probability measure: u l (w s ) = u l (w 0 ) for any w s, so that η,s = η,s = for any s. ν We now briefly study the determinants of the divergence between the physical and the risk neutral probability measure, i.e., we study the determinants of η,s. First, we have dη,s 0 if u ll 0, with a strict inequality if u ll < 0. 3 Intuitively, with respect to the physical probability measure, the risk neutral probability measure overweighs bad states of the world, and underweighs good states of the world. This is illustrated in Figure, which depicts the determinants of η,s, namely u l (w s ) and u l (w 0 ) (cf. equation ()) as a function of w s. Second, if w s > w 0 (respectively w s < w 0 ), then according to the mean value theorem (cf. Simon and Blume (994), p.85) there exists y s (w 0, w s ) (resp. y s (w s, w 0 )) such that: u l (w s ) = u l (w 0 ) + u ll (y s ) (w s w 0 ) (4) Therefore, with u l > 0 and u ll < 0, we have η,s > if and only if w s < w 0. Moreover, given that u l (w s ) > 0 and u l (w 0 ) > 0, () implies that η,s > 0 for any s. Using (3), this in turn implies that risk neutral probabilities are positive for all s. dw s. Risk substitution The variable η,s measures the divergence between the risk neutral probability and the physical probability in state s. We now study its determinants when u ll /= 0, i.e., the risk neutral measure does not coincide with the physical measure. We decompose η,s into several terms to provide an economic interpretation for the change in probability measure in terms of risk substitution. 3 The proof immediately follows from () and the fact that u l is decreasing with u ll < 0, and constant with u ll = 0. 7

8 Figure : The derivation of risk neutral probabilities with u ll < 0 (cf. equations () and (3)). Proposition Let i be the smallest integer such that u (i) = 0 (i 3). Then η,s = + i u (k+) (w ) k= k! 0 (w s w 0 ) k. (5) u l (w 0 ) Proof. With u (i) = 0, we have By definition η,s = u (ws), so that t u t (w0) i u l (w s ) = u l (w 0 ) + u k! (k+) (w 0 )(w s w 0 ) k. k= u (k+) (w ) i η 0,s = + (w s w 0 ) k. (6) k= k! u l (w 0 ) The term u(k+) (w0) is the coefficient of absolute preference for the k + -th degree risk relative u t (w0) to the first degree risk at w 0. A negative preference (i.e., an aversion) for the k + -th degree risk (k+) implies that u (w0) < 0. u t (w0) A first degree risk deterioration is a change in the distribution which is undesirable in the sense of first-order stochastic dominance, i.e., for all agents with increasing utility functions. A second degree risk increase is a change in the distribution which leaves the mean unchanged but is undesirable for all agents with concave utility functions (Rothschild and Stiglitz (970)). Note 8

9 that a first degree risk improvement implies a higher mean, and an increase in second degree risk implies a higher variance at constant mean. A risk neutral investor only has preferences for the first degree risk of the distribution. The change in probability measure described in η,s alters the first degree risk of the distribution of states of the world to incorporate the effect of higher degree risks. With i the smallest integer such that u (i) = 0, the change in probability measure alters the first degree risk to incorporate the effect of second to i th degree risks. This change therefore depends on the relation between the aversion to second to i th degree risks, and the aversion to the first degree risk of the (k+) distribution. This explains why the terms u (w 0), for k + = {,..., i }, appear in (5): Liu where A(w 0 ) u tt (w0) u t (w0) (m) (w0) u t (w0) and Meyer (03) show that ( ) m u for m, is a measure of the rate of substitution between a first degree risk increase and an increase in mth degree risk, i.e., a measure of the willingness to increase a first degree risk to avoid an increase in mth degree risk. In particular, if u (4) = 0, u l (w s ) can be replaced by a second-order Taylor expansion about the point w 0 : u l (w s ) = u l (w 0 ) + u ll (w 0 ) (w s w 0 ) + u lll (w s ) (w s w 0 ). Using this equation and (): u l (w 0 ) + u ll (w 0 ) (w s w 0 ) + u lll (w s ) (w s w 0 ) η,s = (7) u l (w 0 ) = A(w 0 ) (w s w 0 ) + D(w 0 ) (w s w 0 ), (8) is the coefficient of absolute risk aversion at w, D(w ) u ttt (w0) is the u t (w0) 0 0 u t (w0) coefficient of downside risk aversion at w 0 (Modica and Scarsini (005), Crainich and Eeckhoudt (008), Keenan and Snow (00)), and P (w 0 ) u ttt (w 0) is the coefficient of absolute prudence at u tt (w0) w 0 (Kimball (990)). Equation (8) gives a simple decomposition of η,s when the utility function is outer risk neutral it can be risk averse and downside risk averse. With u (4) = 0, the change in probability measure alters the first degree risk to incorporate the effect of the second and third degree risks, also known as risk and downside risk. This change therefore depends on the relation between the aversion to risk and to downside risk, and the aversion to the first degree risk of the distribution. This explains why the coefficients of absolute risk aversion and (m) of downside risk aversion appear in (8): for m =, the term ( ) m u (w 0) mentioned above is u t (w 0 ) equal to A(w 0 ), and for m = 3 it is equal to D(w 0 ). Equation (8) also shows that, with u (4) = 0, downside risk aversion is especially important in explaining the divergence between the risk neutral and the physical probability for levels of future wealth w s that differ substantially from the current level of wealth w 0. More precisely, the effect in the of sense changes that in downside > risk aversion, for w on η,s w dominates >. the effect of changes in risk aversion, dη,s dη,s dd(w0) da(w0) s 0 Finally, when the utility function is downside risk averse but outer risk neutral, i.e., with u lll > 0 and u (4) = 0, equation (8) shows that η,s is quadratic in w s, and is especially large for very high and very low values of w s. Given that η,s is the (scaled) ratio of the risk neutral to the 8

10 λ,s physical probability in state s (η,s ps ν ), this in turn implies that the risk neutral probability distribution has fat tails relative to the physical distribution..3 Asset pricing As has already been shown in many papers and textbooks, the price of any asset can be expressed with risk neutral probabilities. Substituting u l (w s ) from () in (), the price P of an asset with stochastic payoff x may be rewritten as: S P = p s β η,s ul (w 0 ) x s = βν E Λ [x ], (9) u s= l (w 0 ) where E Λ [ ] is the expectation operator with respect to the probability measure Λ. Since this formula must hold for any asset, including the riskfree asset with payoff x s = for all s and with price P f =, by definition of the riskfree rate r f, we have = βν (the expectation of a +r +r f constant under any probability measure is equal to this constant). Substituting in (9): f P = Λ E [x ], (0) + r f which is the standard asset pricing formula with risk neutral probabilities. 3 Downside risk neutral probabilities 3. Definition and properties Assume that u is of class C 3. For any given s, let η 3,s be defined implicitly as (η 3,s exists generically except for u l (w 0 ) + u ll (w 0 )(w s w 0 ) = 0): u l (w s ) η 3,s [u l (w 0 ) + u ll (w 0 )(w s w 0 )]. () We henceforth consider economies such that η 3,s exists for all s. Definition Let ν 3 S s= p s η 3,s, and λ 3,s p sη 3,s ν 3 = S p s η 3,s s= p s η 3,s. () The set {λ 3,s } is the set of downside risk neutral probabilities, and Λ 3 is the downside risk neutral probability measure. By construction, S s= λ 3,s =. Note that η3,s is the Radon-Nikodym derivative of the ν3 downside risk neutral measure with respect to the physical measure. With linear utility or quadratic utility (u lll = 0 in both cases), the downside risk neutral probability measure coincides 9

11 Figure : The derivation of downside risk neutral probabilities with u ll < 0 and u lll > 0 (cf. equations () and ()). with the physical probability measure: by construction, the Taylor expansion u l (w 0 )+u ll (w 0 )(w s w 0 ) is then equal to u l (w s ) for any w s, so that η 3,s = η3,s = for any s. It is important to note that ν3 the downside risk neutral measure is not a risk neutral measure. For example, with quadratic utility, the risk neutral measure would not coincide with the physical measure. We now study the determinants of the divergence between the physical and the downside risk neutral probability measure, as measured by η 3,s, when u lll > 0. The evidence suggests that absolute risk aversion A(w) is nonincreasing in wealth (e.g., Levy (994), Chiappori and Paiella (0)), i.e., it is either constant (CARA) or decreasing (DARA) (note that CRRA utility is DARA). 4 Under this assumption, we have the following relation between η 3,s and future wealth: Claim Suppose that u ll < 0. If the utility function is CARA or DARA and if η 3,s exists, then Proof. Rewrite () as dη 3,s dw s < 0 if w s < w 0. u l (w s ) η 3,s = ul (w ) + u ll (w )(w, (3) w ) 0 0 s 0 4 In the standard version of the portfolio choice problem with a risky asset and a riskfree asset, the dollar amount invested in the risky asset is increasing in wealth if and only if the utility function is DARA (e.g. Gollier (00) p.59). Huang and Stapleton (04) study a similar portfolio choice problem when the investor can also invest in an option, to establish cautiousness as a measure of skewness preference. 0

12 so that dη 3,s dw s u ll (w s )(u l (w 0 ) + u ll (w 0 )(w s w 0 )) u l (w s )u ll (w 0 ) =. (4) (u l (w 0 ) + u ll (w 0 )(w s w 0 )) The denominator in (4) is positive, and the numerator can be rearranged as The sign of A is u l (w 0 )u ll (w s ) u l (w s )u ll (w 0 ) + u ll (w 0 )u ll (w s )(w s w 0 ) (5)./ v./ v A B sign(a) = sign (u ( l (w 0 )u ll (w s ) u l (w s )u ll (w 0 )) (6) u l (w 0 )u ll (w s ) u l (w s )u ll (w 0 ) = sign (7) = sign ( u ll (w s ) u l (w s ) u l (w 0 )u l (w s ) u ll (w 0 ) ul (w 0 ) If the utility function is CARA, then sign(a) = 0. If the utility function is DARA, then sign(a) < 0 if w s w 0 < 0. For u ll < 0 (whether the utility function is CARA or DARA), we have sign(b) < 0 if w s w 0 < 0. To better understand the determinants of the divergence between the downside risk neutral probability and the physical probability, as measured by η 3,s, we now study the difference between u l (w s ) and the term in brackets on the right-hand-side of (). If w s > w 0 (respectively w s < w 0 ), then according to Theorem 30.5 in Simon and Blume (994, p.88) there exists z s (w 0, w s ) (resp. z s (w s, w 0 )) such that: (8) u l (w s ) = u l (w 0 ) + u ll (w 0 ) (w s w 0 ) + u lll (z s ) (w s w 0 ) (9) Therefore, with u l > 0, u ll < 0 and u lll > 0, we have η 3,s > if w s < w 0. This is because u lll > 0 means that u l is convex, so that u l lies above its tangents. However, we do not necessarily have η 3,s > if w s > w 0, because the term in brackets on the right-hand-side of () can then be negative, in which case η 3,s < 0. 5 This is because a first-order Taylor expansion of marginal utility is negative when w s is high enough (in the same way that marginal utility is negative for a high enough argument of the utility function with quadratic utility), so that η 3,s must also be negative for () to hold. This is illustrated in Figure, which depicts the determinants of η 3,s, namely u l (w s ) and u l (w 0 ) + u ll (w 0 )(w s w 0 ) (cf. equation ()) as a function of w s. 5 According to Dirac (94), Negative energies and probabilities should not be considered as nonsense. They are well-defined concepts mathematically, like a negative sum of money, since the equations which express the important properties of energies and probabilities can still be used when they are negative. Like risk neutral probabilities, downside risk neutral probabilities are a mathematical construct. They are not physical probabilities, i.e., they do not represent the probability of occurrence of some events. Instead, their purpose is to provide alternative pricing operators the fact that some of them can be negative is not inherently problematic in that regard.

13 3. Risk substitution The variable η 3,s measures the divergence between the downside risk neutral probability and the physical probability in state s. We now study its determinants when u lll /= 0, i.e., the downside risk neutral measure does not coincide with the physical measure. We decompose η 3,s into several terms to provide an economic interpretation for the change in probability measure in terms of risk substitution. Proposition Let i be the smallest integer such that u (i) = 0 (i 4). Then i ( u l (w ) u ll (w ) 0 0 η 3,s = + +. (0) Proof. With u (i) = 0, we have By definition, k! k= u (k+) (w 0 ) (w s w 0 ) k u (k+) (w 0 ) (w s w 0 ) k i u l (w s ) = u l (w 0 ) + u k! (k+) (w 0 )(w s w 0 ) k. () k= Plugging u l (w s ) from () into () gives (0). η u l (w s ) 3,s = ul (w ) + u ll (w )(w w ). () 0 0 s 0 The term u (k+) (w 0) is the coefficient of absolute preference for the k + -th degree risk relative u t (w0) to the first degree risk at w 0, and u(k+) (w 0) is the coefficient of absolute preference for the k + -th u tt (w0) degree risk relative to the second degree risk at w 0. A downside risk neutral investor only has preferences for the first and second degree risks of the distribution. In what follows, we use the same terminology introduced in section.. The change in probability measure described in η 3,s alters the first and second degree risks of the distribution of states of the world to incorporate the effect of higher degree risks. With i the smallest integer such that u (i) = 0, the change in probability measure alters the first and second degree risks to incorporate the effect of the third to i th degree risks. This change therefore depends on the relation between the aversion to the third to i th degree risks, and the aversion to the first and second degrees of risk of the distribution. This explains why the (k+) 0 (k+) terms u u t (w0) (w ) and u u (w0) tt (w 0 ), for k + = {3,..., i }, appear in (5): Liu and Meyer (03) (m) show that ( ) m u (w0) u t (w0) for m, is a measure of the rate of substitution between an increase in mth degree risk and a first degree risk increase, i.e., a measure of the willingness to increase (m) a first degree risk to avoid an increase in mth degree risk; likewise, ( ) m u (w0) for m, u tt (w0) is a measure of the rate of substitution between an increase in mth degree risk and a second degree risk increase, i.e., a measure of the willingness to increase a second degree risk to avoid an increase in mth degree risk.

14 In particular, if u (4) = 0, then u lll (z s ) = u lll (w 0 ) for any z s, and equation (0) rewrites as ( η 3,s = + D(w 0 ) (w s w 0 ), (3) P (w 0 ) w s w 0 where D(w 0 ) u ttt (w0) is the coefficient of downside risk aversion at w, and P (w ) u ttt (w0) is u t (w0) 0 0 u tt (w0) the coefficient of absolute prudence at w 0. Equation (3) gives as simple decomposition of η 3,s when the utility function is outer risk neutral. With u (4) = 0, the change in probability measure alters the first and second degree risks to incorporate the effect of third degree risk or downside risk. This change therefore depends on the relation between the aversion to downside risk, and the aversion to the first and second degree risks of the distribution. This explains why the coefficients of downside risk aversion and of absolute prudence appear in (3): Liu and Meyer (3) (03) show that ( ) m u (w0) for m =,, is a measure of the rate of substitution between u (m) (w0) an increase in third degree risk and an mth degree risk increase, i.e., a measure of the willingness to increase an mth degree risk to avoid an increase in downside risk. For m =, this term is equal to D(w 0 ), and for m = it is equal to P (w 0 ). 3.3 Asset pricing We now show how to express the price of any asset with downside risk neutral probabilities. Note that the asset price P in Propositions and is the same as the asset price in equations () and (0). With u of class C 3, the price P of an asset with stochastic payoff x may be decomposed in the following terms: Proposition 3 P = E Λ f (w 3 0, w ) x + r f E Λ3 [f(w 0, w )] (4) = E Λ 3 [x ] A(w 0 )cov Λ 3 (w, x ) + r f A(w 0 )E Λ3 [w w 0 ], (5) where f (w 0, w ) is linear in w, and writes as f (w 0, w ) A(w 0 ) [w w 0 ]. Proof. Substituting u l (w s ) from () in () gives S ( P = p s βη 3,s + ull (w 0 ) (ws w 0 ) x s (6) s= u l (w 0 ) u ll (w 0 ) = βν 3 E Λ 3 x + ul (w 0 ) (w w 0 )x = βν 3 E Λ 3 [x ] A(w 0 )E Λ 3 [(w w 0 )x ]. (7) Given that equation (7) must hold for any asset, including the riskfree asset whose payoff is x s = for all s and whose price is by definition of the riskfree rate r f equal to P f =, we +r f have = βν 3 A(w 0 )E Λ 3 [w w 0 ]. + r f 3

15 Substituting in (7) gives P = + r f E Λ 3 [x ] A(w 0)E Λ 3 [(w w 0 )x ] A(w 0 )E Λ3 [w w 0 ]. (8) This formula can be rewritten as in (4) or (5). In equation (5), the utility function only (directly) enters the equation via the coefficient of absolute risk aversion evaluated at the initial level of wealth. Aversion to second degree risk, i.e., risk aversion, is captured by u ll < 0, and it implies A(w 0 ) > 0. Equation (5) shows that an asset whose payoff x is positively correlated with future wealth w under the downside risk neutral measure Λ 3 has a lower price. For preferences such that u lll = 0, and maintaining the assumption that u l (w s ) > 0 for any s (with u ll < 0, this implies that w s is bounded from above), Λ 3 coincides with the physical probability measure, and the utility function can without loss of generality be written as u(w) = w b w, with b 0 if u ll 0. The price of any asset with stochastic payoff x is then 6 A(w P = E [x ] 0 )cov (w, x ) A(w 0 )E [w w 0 ] + rf = + r f E [x ] b cov (w, x ) b E[w ]. (9) In formula (5), the change in probability measure also takes into account the asset price impact of preferences for downside risk and higher degree risks. It is important to note that, in (5), the expression A(w 0 )E Λ 3 [w w 0 ] is strictly positive, as shown in the Supplementary Appendix. We have E Λ 3 [w w 0 ] = 0 if the expected growth in wealth under Λ 3 is nil; E Λ 3 [x ] = 0 if the expected asset payoff under Λ 3 is nil; and cov Λ 3 (w, x ) = 0 if the asset payoff is uncorrelated with aggregate wealth under Λ 3. Mean-variance analysis (with physical probabilities) has at least since Rothschild and Stiglitz (970) been criticized on the grounds that it does not take into account higher-order risk preferences, which leads to substantial pricing errors. Yet, its simplicity and intuitive appeal are such that it remains a cornerstone of finance. We now argue that downside risk neutral probabilities allow to apply mean-variance analysis in asset pricing. In equation (4), the term f (w0,w ) +r f E Λ 3 [f(w0,w )] can be viewed as the pricing kernel associated with downside risk neutral probabilities. Comparing (5) and (9) shows that this pricing kernel corresponds to the one that would obtain with quadratic utility. That is, f (w0,w ) corresponds to the pricing kernel in a world where +r f E[f (w0,w )] agents only have preferences about the mean and the variance of the distribution of their future wealth (with quadratic utility, the expected utility associated with any probability distribution is fully described by its mean and its variance). Thus, using downside risk neutral probabilities allows to price assets in a mean-variance framework. Moreover, equation (4) and the definition of f (w 0, w ) show that the pricing kernel associated with downside risk neutral probabilities is linear in w, in contrast with the stochastic discount factor formula in (). 6 As above, the assumption that u l (w s) > 0 for any s guarantees that A(w 0)E [w w 0] or equivalently be[w ] is strictly positive. 4

16 Comparing the expression in (5) with the one in (0) shows that using downside risk neutral probabilities (the probability measure Λ 3 ) instead of risk neutral probabilities (the probability measure Λ ) results in the apparition of an additional term in the asset pricing formula. Indeed, while the risk neutral probability measure is adjusted to take into account aversion to second and higher degree risks, the downside risk neutral probability measure is only adjusted to take into account aversion to third and higher degree risks. Comparing the expression in (5) with the one in (9) shows that, with downside risk neutral probabilities, assets can be valued as if the representative agent is only averse to first and second degree risks but were neutral with respect to higher degree risks (including downside risk). Higher-order risk preferences such as aversion to downside risk and to outer risk are incorporated in asset prices via a change in the probability measure. As is well-known, a linear pricing kernel allows to use the CAPM to derive the expected return on a security. Denoting by R i x the gross return on a given security i with payoff x, by P Λ 3 R w the gross return on the wealth portfolio with payoff w, and by β Λ 3 cov (Ri,Rw) the security s CAPM beta under the downside risk neutral measure, we have: l E Λ 3 [R i] R f = β Λ 3 IE Λ 3 [R w ] R f i i var Λ 3 (R w ) We refer to the Supplementary appendix for technical details. Crucially, whereas the CAPM with physical probabilities requires strong assumptions, the CAPM that can be derived with downside risk neutral probabilities requires minimal assumptions. (30) 4 Outer risk neutral probabilities 4. Definition and properties Assume that u is of class C 4. For any given s, let η 4,s be defined implicitly as u l (w s ) η 4,s u l (w 0 ) + u ll (w 0 )(w s w 0 ) + u lll (w 0 )(w s w 0 ). (3) Definition 3 Let ν 4 S s= p s η 4,s, and η 4,s p s η 4,s λ 4,s p s = ν 4 S s= p. (3) sη 4,s The set {λ 4,s } is the set of outer risk neutral probabilities, and Λ 4 is the outer risk neutral probability measure. By construction, S λ s= 4,s =. Note that η4,s is the Radon-Nikodym derivative of the outer ν4 risk neutral measure with respect to the physical measure. With linear, quadratic or cubic utility (u (4) = 0 in all three cases), we show below that the outer risk neutral probability measure coincides with the physical probability measure. More generally, with u (4) 0, if w s > w 0 5

17 Figure 3: The derivation of outer risk neutral probabilities with u ll < 0, u lll > 0, and u (4) < 0 (cf. equations (3) and (3)) (respectively w s < w 0 ), then according to Theorem 30.6 in Simon and Blume (994, p.89) there exists ζ s (w 0, w s ) (resp. ζ s (w s, w 0 )) such that: u l (w s ) = u l (w 0 ) + u ll (w 0 ) (w s w 0 ) + u lll (w 0 ) (w s w 0 ) + u (4) 6 Comparing this equality with the one in (3) yields the following result: (ζ s ) (w s w 0 ) 3 (33) Claim With u (4) = 0, η 4,s exists and is equal to for any s. With u (4) < 0 and u ll 0, η 4,s exists and is strictly positive for any s, and η 4,s for w s < w 0. Proof. If u (4) = 0, then the Taylor expansion u l (w 0 ) + u ll (w 0 )(w s w 0 ) + u lll (w 0 )(w s w 0 ) is equal to u l (w s ) (which is strictly positive by assumption) for any w s, so that η 4,s as defined in (3) exists and is equal to for any s. If u (4) < 0 (which given the assumption u ll 0 implies u ll < 0), the expression u l (w 0 ) + u ll (w 0 )(w s w 0 ) + u lll (w 0 )(w s w 0 ), which is in brackets in (3), is quadratic in w s, decreasing in w s at w 0 because u ll < 0, and is greater than u l (w s ) for w s > w 0 because of u (4) < 0 (which can be seen by comparing (3) and (33)). Because u l > 0, this implies that the expression u l (w 0 ) + u ll (w 0 )(w s w 0 ) + u lll (w 0 )(w s w 0 ) in brackets in (3) is strictly positive for any w s, so that η 4,s > 0 for any s. In addition, with u (4) < 0, comparing (3) and (33) shows that η 4,s for w s < w 0. 6

18 The second part of Claim is illustrated in Figure 3, which depicts the determinants of η 4,s, namely u l (w s ) and u l (w 0 ) + u ll (w 0 )(w s w 0 ) + u lll (w 0 )(w s w 0 ) (cf. equation (3)) as a function of w s. 4. Risk substitution The variable η 4,s measures the divergence between the outer risk neutral probability and the physical probability in state s. We now study its determinants when u (4) /= 0, i.e., the outer risk neutral measure does not coincide with the physical measure. We decompose η 4,s into several terms to provide an economic interpretation for the change in probability measure in terms of risk substitution. Proposition 4 Let i be the smallest integer such that u (i) = 0 (i 5), we have i ( u l (w ) u ll (w ) u lll (w ) η 4,s = (34) k! k=3 u (k+) (w 0 ) (w s w 0 ) k u (k+) (w 0 ) (w s w 0 ) k u (k+) (w 0 ) (w s w 0 ) k Proof. With u (i) = 0, we have By definition, η 4,s = i u l (w s ) = u l (w 0 ) + u k! (k+) (w 0 )(w s w 0 ) k. (35) k= Plugging u l (w s ) from (35) into (36) gives (34). u l (w s ). (36) u l (w 0 ) + u ll (w 0 )(w s w 0 ) + u lll (w 0 )(w s w 0 ) The term u(k+) (w0) is the coefficient of absolute preference (or aversion) for the k + -th degree u ttt (w0) risk relative to the third degree risk at w 0. An outer risk neutral investor only has preferences for the first, second and third degree risks of the distribution. In what follows, we use the same terminology introduced in section.. The change in probability measure described in η 4,s alters the first, second and third degree risks of the distribution of states of the world to incorporate the effect of higher degree risks. With i the smallest integer such that u (i) = 0, the change in probability measure alters the first three degrees of risk to incorporate the effect of the fourth to i th degree risks. This change therefore depends on the relation between the aversion to the fourth to i th degree risks, and the aversion to the first three degrees of risk of the distribution. This explains why the terms u (k+) (w 0) u t (w0) u (k+) (w 0) (k+),, and u (w0) u tt (w0), for k + = {4,..., i }, appear in (5): Liu and Meyer (03) m (m) u ttt (w 0 ) show that ( ) ( ) u n u (n) (w 0 ) for m, is a measure of the rate of substitution between an increase in mth degree risk and an increase in nth degree risk, i.e., a measure of the willingness to increase an nth degree risk to avoid an increase in mth degree risk. 7

19 In particular, if u (5) = 0 7 we have u (4) (ζ s ) = u (4) (w 0 ) for any ζ s, and equation (34) rewrites as ( u l (w 0 ) u ll (w 0 ) u lll (w 0 ) η 4,s = + u 6 (4) (w ) (w w ) 3 + u (4) (w ) (w w ) + u (4) (w ) (w w ), (37) 0 s 0 0 s 0 0 s 0 where ( ) m u (4) (w 0), for m =,, 3, are all measures of the intensity of outer risk aversion u (m) (w0) or temperance (Crainich and Eeckhoudt (0), Liu and Meyer (03)). Equation (37) gives a simple decomposition of η 4,s when the utility function is neutral with respect to fifth degree risk it can be risk averse, downside risk averse, and outer risk averse. With u (5) = 0, the change in probability measure alters the first three degrees of risk to incorporate the effect of fourth degree risk or outer risk. This change therefore depends on the relation between the aversion to outer risk, and the aversion to the first three degrees of risk. This explains why several measures of the intensity of aversion to outer risk or temperance appear in (37): Liu and Meyer (03) (4) show that ( ) m u (w0) for m =,, 3, is a measure of the rate of substitution between an u (m) (w0) increase in fourth degree risk and an mth degree risk increase, i.e., a measure of the willingness to increase an mth degree risk to avoid an increase in outer risk. 4.3 Asset pricing We now show how to express the price of any asset with outer risk neutral probabilities. With u of class C 4, the price P of an asset with stochastic payoff x may be decomposed in the following terms: Proposition 5 P = = g(w 0, w ) + r E Λ4 x f E Λ4 [g(w 0, w )] + r f A(w 0 )E Λ4 [w w 0 ] + D(w 0 )E Λ4 [(w w 0 ) ] Λ E Λ 4 [x ] + A(w0)cov (w, x ) + D(w 0 )cov Λ 4 ((w w 0 ), x ), (39) (38) where g(w 0, w ) A(w 0 )[w w 0 ] + D(w 0 )(w w 0 ). Proof. Substituting u l (w s ) from (3) in (), the asset price P may be rewritten as: P = S p s s= ( βη 4,s + ull (w 0 ) u l (w 0 ) (w s w 0 ) + u lll (w 0 ) u l (w 0 ) (w s w 0 ) x s = βν 4 E Λ 4 x A(w 0 )(w w 0 )x + D(w 0 )(w w 0 ) x ( = βν 4 E Λ 4 [x ] A(w 0 )E Λ 4 [(w w 0 )x ] + D(w 0 )E Λ 4 (w w 0 ) x. (40) 7 The approximation of the pricing kernel in Dittmar (00, equation (6)) considers preferences for the first four degrees of risk only, which is consistent with u (5) = 0. 8

20 Given that equation (40) must hold for any asset, including the riskfree asset whose payoff is x s = for all s and whose price is by definition of the riskfree rate r f equal to P f =, we have ( + r = βν 4 A(w 0 )E Λ 4 [w w 0 ] + f D(w 0 )E Λ 4 [(w w 0 ) ]. (4) Substituting (4) in (40) gives +r f E Λ 4 P = [x ] A(w 0)E Λ 4 [(w w 0 )x ] + D(w 0 )E Λ 4 [(w w 0 ) x ] + rf A(w 0 )E Λ4 [w w 0 ] + D(w 0 )E Λ4 [(w w 0 ) ] This formula can be rewritten as in (38) or (39).. (4) The intuition behind equation (39) is that an asset whose payoff x positively covaries with future wealth w will have a lower price if the utility function is risk averse (A(w 0 ) > 0), as in the previous section. In addition, an asset whose payoff tends to be low when future wealth deviates more from its current level will have a lower price if the utility function is averse to downside risk (D(w 0 ) > 0). Consistent with the relation between downside risk and skewness, it is noteworthy that, up to a scaling factor, the term cov ((w w 0 ), x ) can be interpreted as the coskewness of the asset (Harvey and Siddique (000), Chabi-Yo, Leisen, and Renault (04)). In formula (39), the opposite of this covariance measures the contribution of the asset to the downside risk of the wealth portfolio under the outer risk neutral measure (that is, a negative covariance means a positive contribution to downside risk). It is also important to note that, in (39), the expression A(w 0 )E Λ 4 [w w 0 ] + D(w 0 )E Λ 4 [(w w 0 ) ] is strictly positive, as shown in the Supplementary Appendix. As in the previous section, in equation (38), the term g(w0,w ) can be viewed as the +r f E Λ 4 [g(w0,w )] pricing kernel associated with outer risk neutral probabilities. It is quadratic in future wealth according to Proposition 5. We show in the Supplementary Appendix that the expected return on any asset i can be expressed as E Λ 4 [R i] R f = χ cov Λ 4 (R w, R i) + ϑ cov Λ 4 (R, R i), (43) for two constants χ and ϑ. This result is similar to equation (7) in Harvey and Siddique (000), but there is an important difference. Harvey and Siddique (000) use physical probabilities and assume a quadratic stochastic discount factor, so that their result is only an approximation if the stochastic discount factor is not quadratic in wealth. By contrast, using outer risk neutral probabilities ensures that the equation in (43) holds in any case. With u (4) = 0, outer risk neutral probabilities coincide with physical probabilities, and our result coincides with Harvey and Siddique s. With u (4) /= 0, outer risk neutral probabilities will diverge from physical probabilities in such a way that (43) still holds. In the remainder of this section, we study asset pricing when the utility function is outer risk neutral, i.e., u (4) = 0. In this case, the outer risk neutral probability measure Λ 4 coincides with the physical probability measure (cf. Claim ). The formulas in Proposition 5 can then be 9 w

21 applied with physical probabilities. That is, in this case, the price of any asset with stochastic payoff x can simply be expressed as g(w 0, w ) A(w 0 )cov (w, x ) + D(w 0 )cov ((w w 0 ), x ) P = E + r f 0 f (44) x E[g(w, w )] = + r E[x ] + A(w )E[w w ] + D(w )E[(w w ) ]. Note that the expectations and covariances in (44) are computed with physical probabilities. When the utility function is outer risk neutral, the formulas in (44) allow to disentangle between the asset price impact of risk aversion and downside risk aversion. When the utility function is outer risk averse, similar formulas in (38) and (39) apply; the probabilities are then adjusted to take into account the asset price impact of aversion to outer risk and higher degree risks. When the utility function is outer risk neutral, the pricing kernel (with physical probabilities) can be written as g(w0,ws) according to equation (44). It can then be U-shaped: +r f E[g(w0,w )] Claim 3 When the utility function is downside risk averse but outer risk neutral, the pricing kernel is decreasing in wealth for w s < w 0 +, and increasing in wealth for w s > w 0 +. P (w0) Proof. If the utility function is downside risk averse but outer risk neutral, we have u lll > 0 and u (4) = 0. As shown in the Supplementary Appendix, the expression A(w 0 )E Λ 4 [w w 0 ] + D(w 0 )E Λ 4 [(w w 0 ) ], which is equal to E Λ 4 [g(w 0, w )], is strictly positive. Thus, with u (4) = 0, we have E[g(w 0, w )] = E Λ 4 [g(w 0, w )] > 0. Consequently, with + r f > 0, the pricing g(w0,ws) kernel +r f E[g(w 0,w )] is increasing (respectively decreasing) in w s if g(w 0, w s ) is increasing (resp. decreasing) in w s. Given g(w 0, w s ) A(w 0 )[w s w 0 ] + D(w 0 )(w s w 0 ), and with u lll > 0 and only u if l > w 0 which imply D(w 0 ) > 0, the pricing kernel is increasing in future wealth w s if and s > w 0 + A(w0). In addition, A(w0) = u (w0)/u (w0) =. D(w0) D(w0) u ttt (w0)/u t (w0) P (w0) When the utility function is downside risk averse but outer risk neutral, Claim 3 shows how the level of wealth at which the pricing kernel reaches its minimum value is related to risk preferences. The model then predicts that the level of wealth above which the pricing kernel is increasing in wealth is larger than the current level of wealth. Moreover, for a given initial wealth w 0, the higher the coefficient of absolute prudence, the lower the level of future wealth w s above which the pricing kernel is increasing in future wealth. tt Finally, we argue in this paragraph that outer risk neutral probabilities could be viewed as a reasonable approximation of physical probabilities. Indeed, although more research is needed, the empirical evidence is not inconsistent with the hypothesis that agents are prudent but not temperant. It has often been assumed that the preferences for different degrees of risk alternate signs, and indeed they do under CRRA or CARA utility for example. Yet, contrary to what is sometimes believed, there is no clear relation between a preference for a degree of risk and another. For example, Crainich, Eeckhoudt, and Trannoy (03) show that risk lovers can be prudent. The sign of preferences for different degrees of risk is ultimately an empirical question. There is strong direct and indirect empirical evidence for prudence (see Deck and Schlesinger t P (w0) 0